Babuška–Lax–Milgram theorem

In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška, Peter Lax and Arthur Milgram.

In the modern, functional-analytic approach to the study of partial differential equations, one does not attempt to solve a given partial differential equation directly, but by using the structure of the vector space of possible solutions, e.g. a Sobolev space W^k,p. Abstractly, consider two real normed spaces U and V with their continuous dual spaces U^∗ and V^∗ respectively. In many applications, U is the space of possible solutions; given some partial differential operator Λ : U → V^∗ and a specified element f ∈ V^∗, the objective is to find a u ∈ U such that

However, in the weak formulation, this equation is only required to hold when "tested" against all other possible elements of V. This "testing" is accomplished by means of a bilinear function B : U × V → R which encodes the differential operator Λ; a weak solution to the problem is to find a u ∈ U such that

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